This DVR method requires diagonalizing a large \(N \times N\) 
matrix, the dimension of which is given by the product of the
number of spatial collocation points \(N_{\rm c}\)  and the number
of spin basis functions \(N_{\rm s}\) . We use the LAPACK
subroutine DSPEVX to find a selected range of eigenvalues and
eigenvectors [1]}. In order to use a direct
diagonalization procedure to calculate the bound-state
energies [2]} shown in
Refs. [3]}, [4]}, the magnitude of
\(N=N_{\rm c} N_{\rm s}\)  was limited to around 25000 using a
processor with 4 GB of memory. With \(N_{\rm c} \approx 800\) ,
in order to give 5 points per node with about 150 nodes for
threshold wave functions, the number of spin basis functions is
thus restricted to be about \(N_{\rm s} \approx 35\) . When this
is fewer than is needed for a complete calculation, an
approximation scheme becomes necessary, as described in
Section .
